Physics Seminar: Prof Andrew Masters FRSC

‘An old chestnut: can the virial expansion describe the vapour-liquid phase transition?’

by Prof Andrew Masters FRSC,
School of Chemical Engineering & Analytical Science, University of Manchester.

Abstract:

The virial expansion seeks to represent a system property, such as the pressure, as a series expansion in the density. This approach was introduced by Kamerlingh Onnes in 1901 and later put on a firm, theoretical foundation by Mayer and Mayer in 1940. For a classical fluid, where the particles interact via pairwise additive forces, the nth virial coefficient can be written as a sum of integrals involving the pair potential and the temperature. These expressions become extremely complicated for large values of n, but recent technical advances have allowed the numerical evaluation of these coefficients to high order. For example 16 coefficients have recently been calculated for the Lennard-Jones fluid.

The question arises, however, as to whether these virial series converge. For hard bodies, the convergence properties would appear to be excellent. The hard sphere fluid, for example, is well-described up to the freezing transition. Hard spheres, however, do not exhibit a gas-liquid transition. Attractive forces are needed for this. If one considers instead a system like the Lennard-Jones fluid, where there are both repulsions and attractions, convergence issues become more acute. Numerically it seems that above the critical temperature, where there is no gas-liquid transition upon compression, the virial series converges well, possibly up to freezing. Below the critical temperature, however, the series appears to diverge at a density in the region of the vapour density at vapour-liquid co-existence. It has been claimed that this is a fundamental property of the virial series and that it can never describe the gas-liquid transition.

In this talk, I would like to review some of the ideas above and then go on to show that for repulsive fluids, the virial series can predict liquid structure as well as liquid thermodynamic properties. I would then like to present an example for which the virial expansion does indeed predict a vapour-liquid transition, in contradiction to the claim above. Finally I would like to give some thoughts on whether it is possible to carry out a resummation of the virial expansion for the Lennard-Jones fluid so one can describe both the vapour and liquid states with a single function.